MA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "MA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity"

Transcription

1 MA4001 Engineering Mathematics 1 Lecture 10 Limits and Dr. Sarah Mitchell Autumn 2014

2 Infinite limits If f(x) grows arbitrarily large as x a we say that f(x) has an infinite limit. Example: f(x) = 1 x 2. lim x 0 f(x) = The line x = 0 is a vertical asymptote.

3 Example f(x) = 1 x 1 lim x 0+ x = lim 1 x 0 x = 1 Therefore lim does not exist. x 0 x

4 Example f(x) = x x 2 1 Clearly the interesting x values are ±1 and ±. Note that f( x) = f(x), so the function is even. Thus we only need to check the behaviour at 1 and. lim f(x) = + lim x 1+ f(x) = (since x 2 1 < 0 for x < 1) x 1 1+1/x 2 lim f(x) = lim x x 1 1/x 2 = 1 Thus x = ±1 are vertical asymptotes and y = 1 is a horizontal asymptote. Also f(0) = 1.

5 Graph of x x 2 1

6 at a point Definition f(x) is continuous at c, if If c is an interior point of the domain of f lim f(x) = f(c) i.e., the graph of f(x) is continuous at c. x c lim f(x) f(c) or x c lim x c f(x) does not exist then we say that f(x) is discontinuous at c, i.e., the graph is discontinuous at c.

7 Remark If f(x) is not defined at c, then f(x) is neither continuous nor discontinuous at c.

8 Example f(x) = x lim f(x) = x = 2 = f(1) Thus f(x) is continuous at 1. We could repeat this for any value of x, so in fact f(x) is continuous in R.

9 The Heaviside function { 1, x 0 H(x) = is continuous in R\{0}. 0, x < 0 H(x) is discontinuous at x = 0 since lim x 0 H(x) doesn t exist.

10 Example: Removable Singularity f(x) = { x 2 + 1, if x 1 1, if x = 1 f(x) is continuous on R\{1}. f(x) is discontinuous at x = 1 since lim f(x) = lim (x 2 + 1) = = 2 f(1) = 1 x 1 x 1

11 Removable Singularity Definition f(x) has a removable singularity at x = c if f(x) can be made continuous at c by redefining f(c) to be f(c) = lim x c f(x) In the previous example, f(x) had a removable singularity at x = 1. By redefining f(1) = 2 the function is made continuous at x = 1.

12 Example sgn(x) sgn(x) is continuous on its domain R\{0}. At x = 0, sgn(x) is not defined. Thus it is neither continuous nor discontinuous at 0.

13 Left and right continuity Definition f(x) is right continuous at x = c if lim f(x) = f(c). x c+ f(x) is left continuous at x = c if lim f(x) = f(c). x c

14 Example: Heaviside function { 1, x 0 H(x) = is right continuous at x = 0 since 0, x < 0 lim H(x) = 1 = H(0) x 0+ but not left continuous at x = 0 since lim H(x) = 0 1 = H(0) x 0

15 at endpoints of domains Definition f(x) is continuous at a left endpoint of its domain, if it is right continuous at this point. f(x) is continuous at a right endpoint of its domain, if it is left continuous at this point.

16 Example f(x) = 1 x 2 The domain is 1 x 2 0, i.e., x 2 1, i.e., [ 1, 1] f(x) is continuous on ( 1, 1). f(x) is right continuous at x = 1 and left continuous at x = 1. Thus f(x) is continuous on [ 1, 1].

17 Example f(x) = x 2 The domain is x 2 0, i.e., x 2, i.e., [2, ) f(x) is continuous on (2, ). f(x) is right continuous at x = 2 Thus f(x) is continuous on its whole domain [2, ).

18 on R Many functions are continuous on R. These are referred to as continuous functions. Examples: all polynomials; all rational functions with non-zero denominator; sin x, cos x, e x

19 Combinations of continuous functions If f(x) and g(x) are continuous at c then the following are continuous at c: f(x)+g(x) f(x) g(x) f(x)g(x) f(x) g(x) provided g(c) 0.

20 Example sin x, x, e x are all continuous on R. Then 3 sin x e x + 8 x is continuous on its domain R\{0}.

21 of composite functions (f g)(x) = f(g(x)) If g(x) is continuous at c and f(x) is continuous at g(c) then (f g)(x) is continuous at c. Example. f(x) = x, g(x) = x 2 2x + 5. (f g)(x) = x 2 2x + 5 is continuous on R. Note that the domain of f g is R.

22 Continuous functions on [a, b] Recall that f(x) is continuous on the closed interval [a, b] if f(x) is continuous at each x (a, b); f(x) is right continuous at x = a; f(x) is left continuous at x = b.

23 The max-min theorem Theorem If f(x) is continuous on [a, b], there exist x 1, x 2 [a, b], such that f(x 1 ) f(x) f(x 2 ), x [a, b] We say that f(x) has the absolute maximum M = f(x 2 ) at x = x 2 and the absolute minimum m = f(x 1 ) at x = x 1 on [a, b].

24 Corollary If f(x) is continuous on [a, b], then it is bounded on [a, b] i.e., there exists K 0 such that f(x) K for all x [a, b]. Proof. Choose K = max{ f(x 1 ), f(x 2 ) }.

Continuity. DEFINITION 1: A function f is continuous at a number a if. lim

Continuity. DEFINITION 1: A function f is continuous at a number a if. lim Continuity DEFINITION : A function f is continuous at a number a if f(x) = f(a) REMARK: It follows from the definition that f is continuous at a if and only if. f(a) is defined. 2. f(x) and +f(x) exist.

More information

Lecture 5 : Continuous Functions Definition 1 We say the function f is continuous at a number a if

Lecture 5 : Continuous Functions Definition 1 We say the function f is continuous at a number a if Lecture 5 : Continuous Functions Definition We say the function f is continuous at a number a if f(x) = f(a). (i.e. we can make the value of f(x) as close as we like to f(a) by taking x sufficiently close

More information

MTH4100 Calculus I. Lecture notes for Week 8. Thomas Calculus, Sections 4.1 to 4.4. Rainer Klages

MTH4100 Calculus I. Lecture notes for Week 8. Thomas Calculus, Sections 4.1 to 4.4. Rainer Klages MTH4100 Calculus I Lecture notes for Week 8 Thomas Calculus, Sections 4.1 to 4.4 Rainer Klages School of Mathematical Sciences Queen Mary University of London Autumn 2009 Theorem 1 (First Derivative Theorem

More information

1 if 1 x 0 1 if 0 x 1

1 if 1 x 0 1 if 0 x 1 Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

More information

LIMITS AND CONTINUITY

LIMITS AND CONTINUITY LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from

More information

TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

More information

5.1 Derivatives and Graphs

5.1 Derivatives and Graphs 5.1 Derivatives and Graphs What does f say about f? If f (x) > 0 on an interval, then f is INCREASING on that interval. If f (x) < 0 on an interval, then f is DECREASING on that interval. A function has

More information

55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim

55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim Slant Asymptotes If lim x [f(x) (ax + b)] = 0 or lim x [f(x) (ax + b)] = 0, then the line y = ax + b is a slant asymptote to the graph y = f(x). If lim x f(x) (ax + b) = 0, this means that the graph of

More information

Asymptotes. Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if

Asymptotes. Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if Section 2.1: Vertical and Horizontal Asymptotes Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if, lim x a f(x) =, lim x a x a x a f(x) =, or. + + Definition.

More information

Rational Polynomial Functions

Rational Polynomial Functions Rational Polynomial Functions Rational Polynomial Functions and Their Domains Today we discuss rational polynomial functions. A function f(x) is a rational polynomial function if it is the quotient of

More information

Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left.

Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left. Vertical and Horizontal Asymptotes Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left. This graph has a vertical asymptote

More information

1. Then f has a relative maximum at x = c if f(c) f(x) for all values of x in some

1. Then f has a relative maximum at x = c if f(c) f(x) for all values of x in some Section 3.1: First Derivative Test Definition. Let f be a function with domain D. 1. Then f has a relative maximum at x = c if f(c) f(x) for all values of x in some open interval containing c. The number

More information

FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.

FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper. FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is

More information

Math 234 February 28. I.Find all vertical and horizontal asymptotes of the graph of the given function.

Math 234 February 28. I.Find all vertical and horizontal asymptotes of the graph of the given function. Math 234 February 28 I.Find all vertical and horizontal asymptotes of the graph of the given function.. f(x) = /(x 3) x 3 = 0 when x = 3 Vertical Asymptotes: x = 3 H.A.: /(x 3) = 0 /(x 3) = 0 Horizontal

More information

3.5: Issues in Curve Sketching

3.5: Issues in Curve Sketching 3.5: Issues in Curve Sketching Mathematics 3 Lecture 20 Dartmouth College February 17, 2010 Typeset by FoilTEX Example 1 Which of the following are the graphs of a function, its derivative and its second

More information

Notes on Curve Sketching. B. Intercepts: Find the y-intercept (f(0)) and any x-intercepts. Skip finding x-intercepts if f(x) is very complicated.

Notes on Curve Sketching. B. Intercepts: Find the y-intercept (f(0)) and any x-intercepts. Skip finding x-intercepts if f(x) is very complicated. Notes on Curve Sketching The following checklist is a guide to sketching the curve y = f(). A. Domain: Find the domain of f. B. Intercepts: Find the y-intercept (f(0)) and any -intercepts. Skip finding

More information

The Mean Value Theorem

The Mean Value Theorem The Mean Value Theorem THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers

More information

3.4 Limits at Infinity - Asymptotes

3.4 Limits at Infinity - Asymptotes 3.4 Limits at Infinity - Asymptotes Definition 3.3. If f is a function defined on some interval (a, ), then f(x) = L means that values of f(x) are very close to L (keep getting closer to L) as x. The line

More information

Derivatives: rules and applications (Stewart Ch. 3/4) The derivative f (x) of the function f(x):

Derivatives: rules and applications (Stewart Ch. 3/4) The derivative f (x) of the function f(x): Derivatives: rules and applications (Stewart Ch. 3/4) The derivative f (x) of the function f(x): f f(x + h) f(x) (x) = lim h 0 h (for all x for which f is differentiable/ the limit exists) Property:if

More information

APPLICATIONS OF DIFFERENTIATION

APPLICATIONS OF DIFFERENTIATION 4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,

More information

Math 21A Brian Osserman Practice Exam 1 Solutions

Math 21A Brian Osserman Practice Exam 1 Solutions Math 2A Brian Osserman Practice Exam Solutions These solutions are intended to indicate roughly how much you would be expected to write. Comments in [square brackets] are additional and would not be required.

More information

Rolle s Theorem. q( x) = 1

Rolle s Theorem. q( x) = 1 Lecture 1 :The Mean Value Theorem We know that constant functions have derivative zero. Is it possible for a more complicated function to have derivative zero? In this section we will answer this question

More information

Math 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) =

Math 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) = Math 50 Eam # Practice Problem Solutions. Find the vertical asymptotes (if any) of the functions g() = + 4, h() = 4. Answer: The only number not in the domain of g is = 0, so the only place where g could

More information

Curve Sketching GUIDELINES FOR SKETCHING A CURVE: A. Domain. B. Intercepts: x- and y-intercepts.

Curve Sketching GUIDELINES FOR SKETCHING A CURVE: A. Domain. B. Intercepts: x- and y-intercepts. Curve Sketching GUIDELINES FOR SKETCHING A CURVE: A. Domain. B. Intercepts: x- and y-intercepts. C. Symmetry: even (f( x) = f(x)) or odd (f( x) = f(x)) function or neither, periodic function. ( ) ( ) D.

More information

f (x) has an absolute minimum value f (c) at the point x = c in its domain if

f (x) has an absolute minimum value f (c) at the point x = c in its domain if Definitions - Absolute maximum and minimum values f (x) has an absolute maximum value f (c) at the point x = c in its domain if f (x) f (c) holds for every x in the domain of f (x). f (x) has an absolute

More information

Review for Calculus Rational Functions, Logarithms & Exponentials

Review for Calculus Rational Functions, Logarithms & Exponentials Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for

More information

36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?

36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous? 36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this

More information

Fourier Series Chapter 3 of Coleman

Fourier Series Chapter 3 of Coleman Fourier Series Chapter 3 of Coleman Dr. Doreen De eon Math 18, Spring 14 1 Introduction Section 3.1 of Coleman The Fourier series takes its name from Joseph Fourier (1768-183), who made important contributions

More information

3.5 Summary of Curve Sketching

3.5 Summary of Curve Sketching 3.5 Summary of Curve Sketching Follow these steps to sketch the curve. 1. Domain of f() 2. and y intercepts (a) -intercepts occur when f() = 0 (b) y-intercept occurs when = 0 3. Symmetry: Is it even or

More information

Apr 23, 2015. Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23, 2015 sketching 1 / and 19pa

Apr 23, 2015. Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23, 2015 sketching 1 / and 19pa Calculus with Algebra and Trigonometry II Lecture 23 Final Review: Curve sketching and parametric equations Apr 23, 2015 Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23,

More information

Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs

Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s

More information

Polynomial & Rational Functions

Polynomial & Rational Functions 4 Polynomial & Rational Functions 45 Rational Functions A function f is a rational function if there exist polynomial functions p and q, with q not the zero function, such that p(x) q(x) for all x for

More information

Derivatives and Graphs. Review of basic rules: We have already discussed the Power Rule.

Derivatives and Graphs. Review of basic rules: We have already discussed the Power Rule. Derivatives and Graphs Review of basic rules: We have already discussed the Power Rule. Product Rule: If y = f (x)g(x) dy dx = Proof by first principles: Quotient Rule: If y = f (x) g(x) dy dx = Proof,

More information

Learning Objectives for Math 165

Learning Objectives for Math 165 Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given

More information

LIMITS AND CONTINUITY The following tables show values of f(x, y) and g(x, y), correct to three decimal places, for points (x, y) near the origin.

LIMITS AND CONTINUITY The following tables show values of f(x, y) and g(x, y), correct to three decimal places, for points (x, y) near the origin. LIMITS AND Let s compare the behavior of the functions 14. Limits and Continuity In this section, we will learn about: Limits and continuity of various types of functions. sin( ) f ( x, y) and g( x, y)

More information

Differentiability and some of its consequences Definition: A function f : (a, b) R is differentiable at a point x 0 (a, b) if

Differentiability and some of its consequences Definition: A function f : (a, b) R is differentiable at a point x 0 (a, b) if Differentiability and some of its consequences Definition: A function f : (a, b) R is differentiable at a point x 0 (a, b) if f(x)) f(x 0 ) x x 0 exists. If the it exists for all x 0 (a, b) then f is said

More information

f(x) = a x, h(5) = ( 1) 5 1 = 2 2 1

f(x) = a x, h(5) = ( 1) 5 1 = 2 2 1 Exponential Functions an their Derivatives Exponential functions are functions of the form f(x) = a x, where a is a positive constant referre to as the base. The functions f(x) = x, g(x) = e x, an h(x)

More information

Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

More information

Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

More information

MTH 3005 - Calculus I Week 8: Limits at Infinity and Curve Sketching

MTH 3005 - Calculus I Week 8: Limits at Infinity and Curve Sketching MTH 35 - Calculus I Week 8: Limits at Infinity and Curve Sketching Adam Gilbert Northeastern University January 2, 24 Objectives. After reviewing these notes the successful student will be prepared to

More information

Domain and Range. Many problems will ask you to find the domain of a function. What does this mean?

Domain and Range. Many problems will ask you to find the domain of a function. What does this mean? Domain and Range The domain of a function is the set of values that we are allowed to plug into our function. This set is the x values in a function such as f(x). The range of a function is the set of

More information

MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

More information

Section 4.7. Lecture 15. Section 4.7 Vertical and Horizontal Asymptotes; Vertical Tangents and Cusps. Jiwen He

Section 4.7. Lecture 15. Section 4.7 Vertical and Horizontal Asymptotes; Vertical Tangents and Cusps. Jiwen He Section 4.7 Lecture 15 Section 4.7 Vertical and Horizontal Asymptotes; Vertical Tangents and Cusps Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math1431

More information

Contents. 6 Graph Sketching 87. 6.1 Increasing Functions and Decreasing Functions... 87. 6.2 Intervals Monotonically Increasing or Decreasing...

Contents. 6 Graph Sketching 87. 6.1 Increasing Functions and Decreasing Functions... 87. 6.2 Intervals Monotonically Increasing or Decreasing... Contents 6 Graph Sketching 87 6.1 Increasing Functions and Decreasing Functions.......................... 87 6.2 Intervals Monotonically Increasing or Decreasing....................... 88 6.3 Etrema Maima

More information

Math 113 HW #9 Solutions

Math 113 HW #9 Solutions Math 3 HW #9 Solutions 4. 50. Find the absolute maximum and absolute minimum values of on the interval [, 4]. f(x) = x 3 6x 2 + 9x + 2 Answer: First, we find the critical points of f. To do so, take the

More information

D f = (2, ) (x + 1)(x 3) (b) g(x) = x 1 solution: We need the thing inside the root to be greater than or equal to 0. So we set up a sign table.

D f = (2, ) (x + 1)(x 3) (b) g(x) = x 1 solution: We need the thing inside the root to be greater than or equal to 0. So we set up a sign table. . Find the domains of the following functions: (a) f(x) = ln(x ) We need x > 0, or x >. Thus D f = (, ) (x + )(x 3) (b) g(x) = x We need the thing inside the root to be greater than or equal to 0. So we

More information

Review Sheet for Third Midterm Mathematics 1300, Calculus 1

Review Sheet for Third Midterm Mathematics 1300, Calculus 1 Review Sheet for Third Midterm Mathematics 1300, Calculus 1 1. For f(x) = x 3 3x 2 on 1 x 3, find the critical points of f, the inflection points, the values of f at all these points and the endpoints,

More information

Math 181 Spring 2007 HW 1 Corrected

Math 181 Spring 2007 HW 1 Corrected Math 181 Spring 2007 HW 1 Corrected February 1, 2007 Sec. 1.1 # 2 The graphs of f and g are given (see the graph in the book). (a) State the values of f( 4) and g(3). Find 4 on the x-axis (horizontal axis)

More information

MTH 233 Calculus I. Part 1: Introduction, Limits and Continuity

MTH 233 Calculus I. Part 1: Introduction, Limits and Continuity MTH 233 Calculus I Tan, Single Variable Calculus: Early Transcendentals, 1st ed. Part 1: Introduction, Limits and Continuity 0 Preliminaries It is assumed that students are comfortable with the material

More information

Practice with Proofs

Practice with Proofs Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

More information

f(x) = g(x), if x A h(x), if x B.

f(x) = g(x), if x A h(x), if x B. 1. Piecewise Functions By Bryan Carrillo, University of California, Riverside We can create more complicated functions by considering Piece-wise functions. Definition: Piecewise-function. A piecewise-function

More information

So here s the next version of Homework Help!!!

So here s the next version of Homework Help!!! HOMEWORK HELP FOR MATH 52 So here s the next version of Homework Help!!! I am going to assume that no one had any great difficulties with the problems assigned this quarter from 4.3 and 4.4. However, if

More information

Situation: Dividing Linear Expressions

Situation: Dividing Linear Expressions Situation: Dividing Linear Expressions Date last revised: June 4, 203 Michael Ferra, Nicolina Scarpelli, Mary Ellen Graves, and Sydney Roberts Prompt: An Algebra II class has been examining the product

More information

WARM UP EXERCSE. 2-1 Polynomials and Rational Functions

WARM UP EXERCSE. 2-1 Polynomials and Rational Functions WARM UP EXERCSE Roots, zeros, and x-intercepts. x 2! 25 x 2 + 25 x 3! 25x polynomial, f (a) = 0! (x - a)g(x) 1 2-1 Polynomials and Rational Functions Students will learn about: Polynomial functions Behavior

More information

Maths 361 Fourier Series Notes 2

Maths 361 Fourier Series Notes 2 Today s topics: Even and odd functions Real trigonometric Fourier series Section 1. : Odd and even functions Consider a function f : [, ] R. Maths 361 Fourier Series Notes f is odd if f( x) = f(x) for

More information

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

More information

Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)

Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x) SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f

More information

Math 563 Measure Theory Project 1 (Funky Functions Group) Luis Zerón, Sergey Dyachenko

Math 563 Measure Theory Project 1 (Funky Functions Group) Luis Zerón, Sergey Dyachenko Math 563 Measure Theory Project (Funky Functions Group) Luis Zerón, Sergey Dyachenko 34 Let C and C be any two Cantor sets (constructed in Exercise 3) Show that there exists a function F: [,] [,] with

More information

Section 3.7 Rational Functions

Section 3.7 Rational Functions Section 3.7 Rational Functions A rational function is a function of the form where P and Q are polynomials. r(x) = P(x) Q(x) Rational Functions and Asymptotes The domain of a rational function consists

More information

Limits and Continuity

Limits and Continuity Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function

More information

TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

More information

sin(x) < x sin(x) x < tan(x) sin(x) x cos(x) 1 < sin(x) sin(x) 1 < 1 cos(x) 1 cos(x) = 1 cos2 (x) 1 + cos(x) = sin2 (x) 1 < x 2

sin(x) < x sin(x) x < tan(x) sin(x) x cos(x) 1 < sin(x) sin(x) 1 < 1 cos(x) 1 cos(x) = 1 cos2 (x) 1 + cos(x) = sin2 (x) 1 < x 2 . Problem Show that using an ɛ δ proof. sin() lim = 0 Solution: One can see that the following inequalities are true for values close to zero, both positive and negative. This in turn implies that On the

More information

MATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2

MATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2 MATH 10550, EXAM SOLUTIONS (1) Find an equation for the tangent line to at the point (1, ). + y y + = Solution: The equation of a line requires a point and a slope. The problem gives us the point so we

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

More information

correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:

correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were: Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that

More information

Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = e x. y = exp(x) if and only if x = ln(y)

Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = e x. y = exp(x) if and only if x = ln(y) Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = Last day, we saw that the function f(x) = ln x is one-to-one, with domain (, ) and range (, ). We can conclude that f(x) has an inverse function

More information

x a x 2 (1 + x 2 ) n.

x a x 2 (1 + x 2 ) n. Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number

More information

Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

More information

H/wk 13, Solutions to selected problems

H/wk 13, Solutions to selected problems H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.

More information

Pre-Calculus Review Lesson 1 Polynomials and Rational Functions

Pre-Calculus Review Lesson 1 Polynomials and Rational Functions If a and b are real numbers and a < b, then Pre-Calculus Review Lesson 1 Polynomials and Rational Functions For any real number c, a + c < b + c. For any real numbers c and d, if c < d, then a + c < b

More information

Calculus. Contents. Paul Sutcliffe. Office: CM212a.

Calculus. Contents. Paul Sutcliffe. Office: CM212a. Calculus Paul Sutcliffe Office: CM212a. www.maths.dur.ac.uk/~dma0pms/calc/calc.html Books One and several variables calculus, Salas, Hille & Etgen. Calculus, Spivak. Mathematical methods in the physical

More information

Lecture 3: Derivatives and extremes of functions

Lecture 3: Derivatives and extremes of functions Lecture 3: Derivatives and extremes of functions Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2011 Lejla Batina Version: spring 2011 Wiskunde 1 1 / 16

More information

Practice Test - Chapter 1

Practice Test - Chapter 1 Determine whether the given relation represents y as a function of x. 1. y 3 x = 5 2. When x = 1, y = ±. Therefore, the relation is not one-to-one and not a function. The graph passes the Vertical Line

More information

Real Roots of Univariate Polynomials with Real Coefficients

Real Roots of Univariate Polynomials with Real Coefficients Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials

More information

Rational Functions ( )

Rational Functions ( ) Rational Functions A rational function is a function of the form r P Q where P and Q are polynomials. We assume that P() and Q() have no factors in common, and Q() is not the zero polynomial. The domain

More information

The Derivative and the Tangent Line Problem. The Tangent Line Problem

The Derivative and the Tangent Line Problem. The Tangent Line Problem The Derivative and the Tangent Line Problem Calculus grew out of four major problems that European mathematicians were working on during the seventeenth century. 1. The tangent line problem 2. The velocity

More information

CHAPTER 3. Fourier Series

CHAPTER 3. Fourier Series `A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 4 A COLLECTION OF HANDOUTS ON PARTIAL DIFFERENTIAL

More information

Lecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x)

Lecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x) ecture VI Abstract Before learning to solve partial differential equations, it is necessary to know how to approximate arbitrary functions by infinite series, using special families of functions This process

More information

SOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen

SOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen SOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen DEFINITION. A trig inequality is an inequality in standard form: R(x) > 0 (or < 0) that contains one or a few trig functions

More information

Curve Sketching (I) (x + 3)(x + 1)(x 3) 4. 9. Sketch the grapf of the following polynomial function: f(x) = (x + 3)(x 1)(x 2) 4. (x + 1)(x 0)(x 2) 4

Curve Sketching (I) (x + 3)(x + 1)(x 3) 4. 9. Sketch the grapf of the following polynomial function: f(x) = (x + 3)(x 1)(x 2) 4. (x + 1)(x 0)(x 2) 4 1. Sketch the grapf of the following polynomial function: f. Sketch the grapf of the following polynomial function: f. Sketch the grapf of the following polynomial function: f 4. Sketch the grapf of the

More information

Items related to expected use of graphing technology appear in bold italics.

Items related to expected use of graphing technology appear in bold italics. - 1 - Items related to expected use of graphing technology appear in bold italics. Investigating the Graphs of Polynomial Functions determine, through investigation, using graphing calculators or graphing

More information

4.3 Limit of a Sequence: Theorems

4.3 Limit of a Sequence: Theorems 4.3. LIMIT OF A SEQUENCE: THEOREMS 5 4.3 Limit of a Sequence: Theorems These theorems fall in two categories. The first category deals with ways to combine sequences. Like numbers, sequences can be added,

More information

Practice Test - Chapter 1

Practice Test - Chapter 1 Determine whether the given relation represents y as a function of x. 1. y 3 x = 5 When x = 1, y = ±. Therefore, the relation is not one-to-one and not a function. not a function 2. The graph passes the

More information

106 Chapter 5 Curve Sketching. If f(x) has a local extremum at x = a and. THEOREM 5.1.1 Fermat s Theorem f is differentiable at a, then f (a) = 0.

106 Chapter 5 Curve Sketching. If f(x) has a local extremum at x = a and. THEOREM 5.1.1 Fermat s Theorem f is differentiable at a, then f (a) = 0. 5 Curve Sketching Whether we are interested in a function as a purely mathematical object or in connection with some application to the real world, it is often useful to know what the graph of the function

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Math 110 Review for Final Examination 2012 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Match the equation to the correct graph. 1) y = -

More information

Precalculus A 2016 Graphs of Rational Functions

Precalculus A 2016 Graphs of Rational Functions 3-7 Precalculus A 2016 Graphs of Rational Functions Determine the equations of the vertical and horizontal asymptotes, if any, of each function. Graph each function with the asymptotes labeled. 1. ƒ(x)

More information

Homework # 3 Solutions

Homework # 3 Solutions Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8

More information

Section 2.7 One-to-One Functions and Their Inverses

Section 2.7 One-to-One Functions and Their Inverses Section. One-to-One Functions and Their Inverses One-to-One Functions HORIZONTAL LINE TEST: A function is one-to-one if and only if no horizontal line intersects its graph more than once. EXAMPLES: 1.

More information

Problem Set. Problem Set #2. Math 5322, Fall December 3, 2001 ANSWERS

Problem Set. Problem Set #2. Math 5322, Fall December 3, 2001 ANSWERS Problem Set Problem Set #2 Math 5322, Fall 2001 December 3, 2001 ANSWERS i Problem 1. [Problem 18, page 32] Let A P(X) be an algebra, A σ the collection of countable unions of sets in A, and A σδ the collection

More information

Math Rational Functions

Math Rational Functions Rational Functions Math 3 Rational Functions A rational function is the algebraic equivalent of a rational number. Recall that a rational number is one that can be epressed as a ratio of integers: p/q.

More information

100. In general, we can define this as if b x = a then x = log b

100. In general, we can define this as if b x = a then x = log b Exponents and Logarithms Review 1. Solving exponential equations: Solve : a)8 x = 4! x! 3 b)3 x+1 + 9 x = 18 c)3x 3 = 1 3. Recall: Terminology of Logarithms If 10 x = 100 then of course, x =. However,

More information

PRE-CALCULUS GRADE 12

PRE-CALCULUS GRADE 12 PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.

More information

AP Calculus AB Home Enjoyment Ch. 3A 89 Curve Sketching

AP Calculus AB Home Enjoyment Ch. 3A 89 Curve Sketching AP Calculus AB Home Enjoyment Ch. 3A 89 Curve Sketching Date Day Objective Calculus Assignments 10/1/2015 Thursday Absolute Extrema, Extreme Value Theorem Ch. 3A HW #1 10/2/2015 Friday Fair Day Fair Day

More information

Chapter 2 Limits Functions and Sequences sequence sequence Example

Chapter 2 Limits Functions and Sequences sequence sequence Example Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students

More information

Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) =

Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) = Vertical Asymptotes Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: lim f (x) = x a lim f (x) = lim x a lim f (x) = x a

More information

Functions: Piecewise, Even and Odd.

Functions: Piecewise, Even and Odd. Functions: Piecewise, Even and Odd. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway September 21-22, 2015 Tutorials, Online Homework.

More information

5.3 Improper Integrals Involving Rational and Exponential Functions

5.3 Improper Integrals Involving Rational and Exponential Functions Section 5.3 Improper Integrals Involving Rational and Exponential Functions 99.. 3. 4. dθ +a cos θ =, < a

More information

Slant asymptote. This means a diagonal line y = mx + b which is approached by a graph y = f(x). For example, consider the function: 2x 2 8x.

Slant asymptote. This means a diagonal line y = mx + b which is approached by a graph y = f(x). For example, consider the function: 2x 2 8x. Math 32 Curve Sketching Stewart 3.5 Man vs machine. In this section, we learn methods of drawing graphs by hand. The computer can do this much better simply by plotting many points, so why bother with

More information

Prompt Students are studying multiplying binomials (factoring and roots) ax + b and cx + d. A student asks What if we divide instead of multiply?

Prompt Students are studying multiplying binomials (factoring and roots) ax + b and cx + d. A student asks What if we divide instead of multiply? Prompt Students are studying multiplying binomials (factoring and roots) ax + b and cx + d. A student asks What if we divide instead of multiply? Commentary In our foci, we are assuming that we have a

More information

Draft Material. Determine the derivatives of polynomial functions by simplifying the algebraic expression lim h and then

Draft Material. Determine the derivatives of polynomial functions by simplifying the algebraic expression lim h and then CHAPTER : DERIVATIVES Specific Expectations Addressed in the Chapter Generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function,

More information